The Binet-Legendre Metric in Finsler Geometry
The Binet-Legendre Metric in Finsler Geometry
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DOI：
10.2140/gt.2012.16.2135
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发表时间：
2011-04
影响因子：
2
通讯作者：
V. Matveev;M. Troyanov
中科院分区：
文献类型：
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作者：
V. Matveev;M. Troyanov
对于每一个Finsler度量F，我们都有一个黎曼度量GF(称为比内-勒让德度量)。黎曼度量GF在Finsler度量F的共形变形下表现得很好，这使它成为Finsler几何中的一个强有力的工具。我们通过解决一些命名的Finslerian几何问题来说明这一点。我们还推广并给出了一些已知结果的新的、更短的证明。特别地，我们回答了M MatSumoto关于两个Minkowski空间之间的局部共形映射的一个问题，我们刻画了Finsler流形上所有可能的共形自映射和所有自相似。我们还对所有紧致共形平坦Finsler流形进行了分类，解决了S、邓、侯关于局部对称Finsler空间的Berwaldian特征的一个猜想，并将王海昌关于Finsler流形的等距群的最大维数的一个经典结果推广到了所有维的流形上。本文中的大多数证明都遵循以下方案：使用对应的F7！我们将Finslerian问题归结为一个类似的问题，这是比内-勒让德度量更容易的问题，并且在我们所考虑的大多数情况下已经得到了解决。黎曼问题的解为我们提供了有助于解决初始芬斯勒问题的附加信息。我们的方法即使在没有Finsler几何中通常假定的强凸性假设的情况下也适用。光滑性假设也可以用较弱的部分光滑性代替，这是我们在文中引入的一个概念。因此，我们的结果适用于芬斯勒文献中通常不考虑的一大类芬斯勒指标。
For every Finsler metric F we associate a Riemannian metric gF (called the Binet‐ Legendre metric). The Riemannian metric gF behaves nicely under conformal deformation of the Finsler metric F , which makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, we solve a conjecture of S Deng and Z Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend a classic result by H C Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence F 7! gF we reduce the Finslerian problem to a similar problem for the Binet‐ Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by a weaker partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.